Parallel & Perpendicular Lines
Algebra, Lines
What are parallel and perpendicular lines
Definition: Parallel lines have the same slope. Perpendicular lines meet at a right angle and their slopes multiply to -1 when both slopes are defined.
Formula: $$m_{\parallel}=m,\quad m_{\perp}=-\frac{1}{m}$$
Intro: This perpendicular line calculator finds the equation of the line parallel or perpendicular to your given line through your point. It also works as a perpendicular gradient calculator.
Accepted line forms
- Slope–intercept: $y=mx+b$
- Standard form: $Ax+By=C$ ( $m=-A/B$ )
- Vertical line: $x=c$ ( $perpendicular\;is\;y=\text{constant}$ )
How this calculator works
- Enter your given line in a common form like
y = (2/3)x - 1or2x + 3y = 6. - Enter the point
(x0, y0). - Choose Parallel or Perpendicular.
- MathGPT extracts the slope, compute the target slope, then build the line through your point using point slope form.
Worked example
- Line $y=\tfrac{2}{3}x - 1$; find the perpendicular through $(3,4)$.
- Identify slope of the given line in $y=mx+b$ form: $$m=\tfrac{2}{3}.$$
- Perpendicular slope is the negative reciprocal: $$m_{\perp}=-\frac{1}{m}=-\frac{1}{\tfrac{2}{3}}=-\tfrac{3}{2}.$$
- Use point–slope form with point $(x_0,y_0)=(3,4)$: $$y-y_0=m_{\perp}(x-x_0)\;\Rightarrow\; y-4=-\tfrac{3}{2}(x-3).$$
- Distribute and solve for $y$: $$y-4=-\tfrac{3}{2}x+\tfrac{9}{2} \;\Rightarrow\; y=-\tfrac{3}{2}x+\tfrac{9}{2}+4.$$
- Combine constants (write $4=\tfrac{8}{2}$): $$y=-\tfrac{3}{2}x+\tfrac{9}{2}+\tfrac{8}{2}=-\tfrac{3}{2}x+\tfrac{17}{2}.$$
- Final perpendicular line through $(3,4)$: $$\boxed{y=-\tfrac{3}{2}x+\tfrac{17}{2}}.$$
- Quick check: Slope product $m\cdot m_{\perp}=\tfrac{2}{3}\cdot(-\tfrac{3}{2})=-1$ ✔️; point check: plug $x=3$ gives $y=-\tfrac{9}{2}+\tfrac{17}{2}=\tfrac{8}{2}=4$ ✔️.
- Line $y=-\tfrac{5}{4}x+7$; find the parallel line through $(−2,1)$.
- Given slope: $$m=-\tfrac{5}{4}.$$
- Parallel lines have the **same** slope: $$m_{\parallel}=m=-\tfrac{5}{4}.$$
- Point–slope with $(x_0,y_0)=(-2,1)$: $$y-1=-\tfrac{5}{4}(x-(-2))=-\tfrac{5}{4}(x+2).$$
- Expand and solve for $y$: $$y-1=-\tfrac{5}{4}x-\tfrac{5}{2}\;\Rightarrow\; y=-\tfrac{5}{4}x-\tfrac{5}{2}+1.$$
- Combine constants (write $1=\tfrac{2}{2}$): $$y=-\tfrac{5}{4}x-\tfrac{5}{2}+\tfrac{2}{2}=-\tfrac{5}{4}x-\tfrac{3}{2}.$$
- Final parallel line through $(-2,1)$: $$\boxed{y=-\tfrac{5}{4}x-\tfrac{3}{2}}.$$
- Check the point: $x=-2 \Rightarrow y=-\tfrac{5}{4}(-2)-\tfrac{3}{2}=\tfrac{10}{4}-\tfrac{3}{2}=\tfrac{5}{2}-\tfrac{3}{2}=1$ ✔️.
- Line $x=4$; find a perpendicular through $(3,4)$.
- A vertical line $x=4$ has undefined slope.
- Perpendicular to a vertical line is **horizontal**: slope $0$, equation $y=\text{constant}$.
- Horizontal line through $(3,4)$ is $$\boxed{y=4}.$$
- Parallel to $x=4$ through $(3,4)$ would be another vertical line: $$\boxed{x=3}.$$
FAQs
What if the given line is vertical?
Parallel is another vertical line $x=c$; perpendicular is horizontal $y=c$.
Do I need slope–intercept form first?
If the line is not already in $y=mx+b$, rearrange to isolate $y$ and read $m$. For vertical lines ($x=c$), treat separately as above.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
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More ways MathGPT can help
- Ask questions and get step by step explanations you can copy into your notes.
- Practice mode to generate similar parallel and perpendicular line problems.
- Create flashcards from the rules and examples. ( Flashcard )
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